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MA 15400 Section 6.2 The Fundamental Identities: (1) The reciprocal identities: 1 1 1 csc sec cot sin cos tan Note: (sin ) 2 is usually written sin . Similar notation for the other functions.) 2 Lesson 3 Trigonometric Function of Angles (2) The tangent and cotangent identities: sin cos tan cot cos sin (3) The Pythagorean identities: sin2 cos 2 1 1 tan2 sec2 1 cot 2 csc2 The reciprocal identities are obvious from the definitions of the six trigonometric functions. Take the simple right triangle with sides 3, 4 and 5 with opposite the side of length 3. 5 3 θ 4 To prove the tangent identity, examine the following. The cotangent identity proof is similar. opp opp opp hyp hyp sin tan adj hyp adj adj cos hyp 3 4 3 cos tan 5 5 4 3 sin 5 3 5 3 cos 4 5 4 4 5 sin tan cos sin 2 2 Find sin and cos, now find sin cos 3 4 cos 5 5 2 2 (sin ) (cos ) ? sin sin 2 cos 2 ? 2 2 3 4 ? 5 5 9 16 ? 25 25 25 1 25 sin 2 cos 2 1 r θ y x x2 y 2 r 2 opp 2 adj 2 hyp 2 x2 y 2 r 2 r2 r2 r2 opp 2 adj 2 hyp 2 hyp 2 hyp 2 hyp 2 cos 2 sin 2 1 sin 2 cos 2 1 sin2 cos 2 1 is a Pythagorean identity since it is derived from the Pythagorean Theorem. 1 MA 15400 Lesson 3 Trigonometric Function of Angles Section 6.2 Divide both sides by sin2 to find another Pythagorean identity. sin 2 cos 2 1 Divide each side by sin 2 sin 2 cos 2 1 2 2 sin sin sin 2 cos 2 1 csc sin 2 1 cot csc 2 2 The third Pythagorean identity can by found by dividing the original sin 2 cos2 1 by cos2 . Each of the three Pythagorean identities creates two more identities by subtracting a term from the left side to the right side. sin 2 cos 2 1 1 tan 2 sec 2 1 cot 2 csc 2 sin 2 1 cos 2 tan 2 sec 2 1 cot 2 csc 2 1 cos 2 1 sin 2 1 sec 2 tan 2 1 csc 2 cot 2 Verify the identity by transforming the left side into the right side. tan cot 1 sin3 cot 3 cos3 sec csc tan sin cos 2 2 1 cot 2 sin 2 2 MA 15400 Lesson 3 Trigonometric Function of Angles Section 6.2 (1cos )(1 cos ) sin cos sec 1 sin 1sin 1 tan 1 cot tan csc sec 2 2 2 2 2 2 3 MA 15400 Lesson 3 Trigonometric Function of Angles Section 6.2 Using the coordinate system, draw an angle θ in standard position (vertex at the origin and the x-axis is the initial side). y = opp Notice that the adjacent side corresponds to the x-value of the coordinate and the opposite side corresponds the y-value of the coordinate. (x, y) The idea that the cosine of corresponds to the x-axis and the sine of corresponds to the y-axis is one that you need to get used to. This is not saying that sin θ = the y value nor that cos θ = the x value. It simply says there is a correspondence. x = adj If is an angle in standard position on a rectangular coordinate system and if P(-5, 12) is on the terminal side of , find the values of the six trigonometric functions of . If is an angle in standard position on a rectangular coordinate system and if P(4, 3) is on the terminal side of , find the values of the six trigonometric functions of . 4 MA 15400 Lesson 3 Trigonometric Function of Angles Section 6.2 Find the exact values of the six trigonometric functions of , if is in standard position and the terminal side of is in the specified quadrant and satisfies the given condition. III; on the line 4x – 3y = 0 II; parallel to the line 3x + y – 7 = 0 Notice: tan θ = slope of the line! Find the quadrant containing if the given conditions are true. a) tan < 0 and cos > 0 b) sec > 0 and tan < 0 c) csc > 0 and cot < 0 d) cos < 0 and csc < 0 e) cos θ < 0 and sec θ > 0 II sin + csc tan III cot + I All + cos IV sec + To help you remember the picture above: Think (in order of quadrants): ALL STUDENTS TAKE CALCULUS. Quadrant I: All functions are positive values. Quadrant II: Sine and its inverse are positive, others negative. Quadrant III: Tangent and its inverse are positive, others negative. Quadrant IV: Cosine and its inverse are positive, other negative. 5 MA 15400 Lesson 3 Trigonometric Function of Angles Section 6.2 Use the fundamental identities to find the values of the trigonometric functions for the given conditions: tan 12 and cos 0 5 sec 4 and csc 0 6